Given: \(\mathrm{TC}=\mathrm{FC}+\mathrm{VC}\) prove that: \(\quad \mathrm{ATC}=\mathrm{AFC}+\mathrm{AVC}\)

Let's begin with our given equation: \(\mathrm{TC} = \mathrm{FC} + \mathrm{VC}\) Here, Total Cost (TC) is comprised of Fixed Costs (FC) and Variable Costs (VC). Now we need to find ATC, AFC, and AVC, the average versions of these costs with respect to output.

First, let's find the Average Total Cost (ATC). We calculate ATC by dividing the Total Cost (TC) by the output, which we represent as Q: \(\mathrm{ATC} = \frac{\mathrm{TC}}{Q}\) Next, we find the Average Fixed Cost (AFC) by dividing the Fixed Costs (FC) by the output (Q): \(\mathrm{AFC} = \frac{\mathrm{FC}}{Q}\) Finally, we find the Average Variable Cost (AVC) by dividing the Variable Costs (VC) by the output (Q): \(\mathrm{AVC} = \frac{\mathrm{VC}}{Q}\) Now that we have the average cost formulas, let's move on to the proof.

We know that: \(\mathrm{TC}=\mathrm{FC}+\mathrm{VC}\) Now, divide both sides of the equation by Q: \(\frac{\mathrm{TC}}{Q} = \frac{\mathrm{FC} + \mathrm{VC}}{Q}\) Now, we can separate the terms on the right side: \(\frac{\mathrm{TC}}{Q} = \frac{\mathrm{FC}}{Q} + \frac{\mathrm{VC}}{Q}\) Now, recall the formulas we derived for ATC, AFC, and AVC: \(\mathrm{ATC} = \frac{\mathrm{TC}}{Q}\) \(\mathrm{AFC} = \frac{\mathrm{FC}}{Q}\) \(\mathrm{AVC} = \frac{\mathrm{VC}}{Q}\) Replace the terms in the equation with these cost formulas: \(\mathrm{ATC} = \mathrm{AFC} + \mathrm{AVC}\) Using this, we have now successfully proven that the average costs add up as expected: \(\mathrm{ATC} = \mathrm{AFC} + \mathrm{AVC}\)